Question

Which of the following properties of Matrix \(A = \begin{bmatrix} 1 & 0.5 & 0 \\ 0.5 & 1 & 0.5 \\ 0 & 0.5 & 1 \end{bmatrix}\) are CORRECT?

• A. Singular
• B. Positive definite
• C. Symmetric
• D. Diagonal

Hint:

A symmetric matrix has the same eigenvalues, and it is positive definite if all its eigenvalues are positive. A matrix with non-zero off-diagonal elements is not diagonal.

Answer 1

The Correct Option is B, C

Step 1: Understand the matrix properties.
A matrix is said to be symmetric if its transpose is equal to the matrix itself. A matrix is positive definite if all its eigenvalues are positive. A matrix is diagonal if all its off-diagonal elements are zero. A matrix is singular if its determinant is zero.
Step 2: Analyze the properties of the given matrix.
- The matrix \(A\) is symmetric because \(A = A^T\).
- To check if it is positive definite, we can check its eigenvalues. Since the matrix has positive eigenvalues, it is positive definite.
- The matrix is not diagonal because it has non-zero off-diagonal elements.
- The matrix is not singular because its determinant is non-zero.
Thus, the correct answers are (2) Positive definite and (3) Symmetric.